Quantum Channels and Completely Positive Trace-Preserving Maps

Quantum channels are the daily language of experimental quantum computing. When a group reports a two-qubit gate fidelity, they have run quantum process tomography or gate-set tomography: prepare a spanning set of inputs, measure the outputs, and fit a Choi matrix subject to $J\ge 0$ and $\mathrm{Tr}_B J=I$ — a semidefinite program that is Choi's theorem in action. The Choi matrix, or its diamond-norm distance to the ideal gate, is the number gates on superconducting, trapped-ion, and photonic platforms are benchmarked against; randomized benchmarking is the scalable shortcut. In error correction, the depolarizing and amplitude/phase-damping channels of this article are the standard noise models whose thresholds decide whether fault tolerance is possible at all. In communication, the complementary channel and the coherent information $I_c=S(\mathcal{E}(\rho))-S(\mathcal{E}^c(\rho))$ set the quantum capacity, whose non-additivity remains open research. And the transpose's failure of complete positivity is no curiosity — positivity of the partial transpose is the Peres–Horodecki criterion, one of the most-used entanglement tests in the lab. Wherever a real device meets a decohering world, a CPTP map is the bookkeeping.

Every channel is the same story: unitary on a bigger box, then close your eyes to part of it. Adjoin to your system $A$ an environment $E$ in a fixed pure state $|0\rangle_E$, evolve the pair under a single unitary $U$, then throw the environment away by a partial trace. Whatever irreversible, decohering thing happens to $A$ is just the shadow of a reversible evolution on a space you cannot fully see. Stinespring's theorem promotes this from example to law: every channel arises this way, so "open-system dynamics" and "unitary-plus-forgetting" are the same set of maps.

Why demand complete positivity instead of the seemingly sufficient "sends states to states"? Because your qubit may not be alone. Suppose $A$ is one half of a Bell pair whose partner $R$ sits in a colleague's lab across the country. Whatever the shared correlations, a physical operation on $A$ must leave the joint $AR$ description a legitimate state — no negative probabilities. Positivity of $\mathcal{E}$ alone does not guarantee this; positivity of $\mathcal{E}\otimes\mathrm{id}_R$ does. The transpose is the cautionary tale: it maps every single-qubit state to a valid state, yet applied to half of a Bell pair it yields an operator with a negative eigenvalue — a phantom "state" assigning negative probability to an outcome. Complete positivity is the demand that the map stay sane before any silent spectator. The Choi matrix turns all of this into linear algebra. Feed half of an unnormalized maximally entangled vector through the channel and out comes one bipartite operator $J(\mathcal{E})$ that fingerprints the whole map: its positivity is complete positivity, its output partial trace equalling the identity is trace preservation, and its rank counts the minimal Kraus operators. Abstract questions about superoperators become concrete questions about the eigenvalues of one matrix — and the next figure runs exactly that test on a real channel and on the transpose counterexample.

Finally, channels compose: $\mathcal{E}_2\circ\mathcal{E}_1$ is again CPTP. This monoid structure, with the identity channel as unit, is why information can only degrade along a processing chain — every extra channel is another chance to lose distinguishability, never to regain it. That is the data-processing inequality, and why "post-processing cannot help" is a theorem rather than a slogan.

A channel is fixed by two conditions: it preserves the probabilistic normalization and keeps every extended state legitimate. The first line states the state-to-state consequence; the deeper condition, complete positivity, gets its own equation next.

where: $\rho$ — input density operator on $\mathcal{H}_A$; $\mathcal{E}$ — linear superoperator $\mathcal{L}(\mathcal{H}_A)\to\mathcal{L}(\mathcal{H}_B)$; $\ge 0$ — positive semidefinite. Reading: valid states go to valid states, but this single-system condition is necessary and not sufficient. Complete positivity demands the map stay positive even acting on a subsystem of a larger, possibly entangled, whole — and one need only test a reference $R$ as large as the input.

where: $\mathrm{id}_R$ — do-nothing channel on a spectator $R$; $\Omega$ — any positive operator on $\mathcal{H}_A\otimes\mathcal{H}_R$. Reading: positivity must survive a silent bystander; the transpose fails here despite passing (1), which is exactly why CP is the right notion. The Kraus or operator-sum form is the computational workhorse: a finite sum of "sandwich" terms whose completeness relation is trace preservation.

where: $E_k$ — Kraus operators $\mathcal{H}_A\to\mathcal{H}_B$, at most $d_A d_B$ of them; $\sum_k E_k^\dagger E_k=I$ — completeness, equivalent to $\mathrm{Tr}\,\mathcal{E}(\rho)=\mathrm{Tr}\,\rho$. Reading: summing the branches $E_k$ incoherently is what makes the evolution non-unitary. Stinespring's dilation is the physical origin of the Kraus form — an isometry into a larger space, then discard the added factor. It is the equation behind the first figure.

where: $V$ — isometry $\mathcal{H}_A\to\mathcal{H}_B\otimes\mathcal{H}_E$; $\mathrm{Tr}_E$ — partial trace over the environment. Reading: $V^\dagger V=I_A$ (not $VV^\dagger=I$) embeds $A$ into a corner of the bigger space; forgetting $E$ injects the noise. The Choi–Jamiołkowski isomorphism folds the whole channel into one operator on the doubled space by running half of a maximally entangled vector through it.

where: $J(\mathcal{E})$ — Choi operator on $\mathcal{H}_B\otimes\mathcal{H}_A$; $|\Gamma\rangle$ — unnormalized maximally entangled vector of norm $\sqrt{d_A}$. Reading: $\mathcal{E}\mapsto J(\mathcal{E})$ is a bijection — "channel–state duality" — so channel questions become state questions. The payoff is a concrete validity test: complete positivity becomes a matrix being positive, trace preservation becomes a partial-trace condition.

where: $J(\mathcal{E})\ge 0$ — equivalent to complete positivity; $\mathrm{Tr}_B J(\mathcal{E})=I_A$ — equivalent to trace preservation. Reading: two checks on one matrix replace the "for all reference systems" quantifier of (2). The inverse isomorphism reconstructs the channel's action from its Choi operator, closing the dictionary.

where: $\rho^{\mathsf T}$ — transpose of the input in the $\{|i\rangle\}$ basis; $\mathrm{Tr}_A$ — partial trace over the input-copy register. Reading: channel and Choi matrix carry identical information. Because every channel is an isometry plus a partial trace, and neither raises distinguishability, relative entropy can only fall — the master monotonicity of the theory.

where: $D(\rho\Vert\sigma)=\mathrm{Tr}[\rho(\log\rho-\log\sigma)]$ — quantum relative entropy; $\mathcal{E}$ — any CPTP map on both arguments. Reading: no processing makes two states easier to tell apart — the data-processing inequality, parent of the Holevo bound and capacity converses. The dilation defines a second channel for free: keep the environment instead of the output. That complementary channel governs how much information leaks away.

where: $\mathcal{E}^c$ — complementary channel $\mathcal{H}_A\to\mathcal{H}_E$; $\mathrm{Tr}_B$ — trace out the output instead of the environment. Reading: $\mathcal{E}$ and $\mathcal{E}^c$ are two halves of one isometry, and their interplay (degradability) fixes the quantum capacity.

The CPTP definition contains every familiar kind of evolution as a special case — from perfectly reversible to maximally destructive, down to ordinary classical noise.

1 · The amplitude-damping channel is CPTP — proved by its Choi matrix (fully worked)

The amplitude-damping channel models energy relaxation ($T_1$ decay): an excited qubit emits a photon into the environment with probability $\gamma$. Its Kraus operators are $E_0=|0\rangle\langle 0|+\sqrt{1-\gamma}\,|1\rangle\langle 1|$ and $E_1=\sqrt{\gamma}\,|0\rangle\langle 1|$. Show it is a valid channel, and verify complete positivity numerically at $\gamma=0.2$.

Step 1. Check the completeness relation (trace preservation). Compute $E_0^\dagger E_0=|0\rangle\langle 0|+(1-\gamma)|1\rangle\langle 1|$ and $E_1^\dagger E_1=\gamma|1\rangle\langle 1|$. Their sum is $|0\rangle\langle 0|+(1-\gamma+\gamma)|1\rangle\langle 1|=|0\rangle\langle 0|+|1\rangle\langle 1|=I$. So $\sum_k E_k^\dagger E_k=I$ and the channel is trace-preserving. The Kraus set is complete.

Step 2. Build the Choi matrix $J=\sum_{i,j}\mathcal{E}(|i\rangle\langle j|)\otimes|i\rangle\langle j|$. The Kraus operators give $\mathcal{E}(|0\rangle\langle 0|)=|0\rangle\langle 0|$, $\mathcal{E}(|1\rangle\langle 1|)=\gamma|0\rangle\langle 0|+(1-\gamma)|1\rangle\langle 1|$, $\mathcal{E}(|0\rangle\langle 1|)=\sqrt{1-\gamma}\,|0\rangle\langle 1|$. In the basis $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ (first index output $B$) the nonzero entries are $J_{00,00}=1$, $J_{01,01}=\gamma$, $J_{11,11}=1-\gamma$, $J_{00,11}=J_{11,00}=\sqrt{1-\gamma}$.

Step 3. Diagonalize. The $\{|00\rangle,|11\rangle\}$ subspace decouples into the $2\times 2$ block with diagonal entries $1$ and $1-\gamma$ and off-diagonal entry $\sqrt{1-\gamma}$; it has trace $2-\gamma$ and determinant $(1)(1-\gamma)-(1-\gamma)=0$, so its eigenvalues are $2-\gamma$ and $0$. The remaining diagonal entries contribute $\gamma$ and $0$. Hence the four Choi eigenvalues are $\{2-\gamma,\ \gamma,\ 0,\ 0\}$, all $\ge 0$ for $0\le\gamma\le 1$. The channel is completely positive.

Step 4. Insert $\gamma=0.2$: the eigenvalues are $\{1.8,\ 0.2,\ 0,\ 0\}$ — exactly the green bars in the second figure. The Choi rank is $2$, matching the two Kraus operators, and $\mathrm{Tr}_B J=I_A$ confirms trace preservation again. At $\gamma=0.2$ the Choi spectrum is $\{1.8, 0.2, 0, 0\}\ge 0$, so amplitude damping is CPTP.

2 · The transpose map fails — the canonical counterexample (fully worked)

Show that the transpose $T(\rho)=\rho^{\mathsf T}$, though positive and trace-preserving on a single qubit, is not a quantum channel, by computing what it does to half of a Bell state and by inspecting its Choi operator.

Step 1. Take the Bell state $|\Phi^+\rangle=\tfrac{1}{\sqrt 2}(|00\rangle+|11\rangle)$, with density matrix having entries $\tfrac12$ at positions $00,00$; $00,11$; $11,00$; $11,11$. Apply $T$ to the second qubit only (the partial transpose), which swaps the $01,10$ and $10,01$ index pairs. The result is the partial transpose $(\Phi^+)^{T_B}$.

Step 2. Diagonalize the partially transposed matrix. Its eigenvalues are $\{\tfrac12,\ \tfrac12,\ \tfrac12,\ -\tfrac12\}$ — three positive and one negative. A genuine density operator cannot have a negative eigenvalue, so $T\otimes\mathrm{id}$ has produced a non-state. The $-\tfrac12$ eigenvalue is the smoking gun: $T$ is not completely positive.

Step 3. Confirm via the Choi operator. By definition $J(T)=\sum_{ij}T(|i\rangle\langle j|)\otimes|i\rangle\langle j|=\sum_{ij}|j\rangle\langle i|\otimes|i\rangle\langle j|$, precisely the SWAP operator — eigenvalue $+1$ on the symmetric subspace and $-1$ on the antisymmetric one, spectrum $\{+1,+1,+1,-1\}$ (the blue-and-red bars in the second figure). Since $J(T)\not\ge 0$, the Choi criterion (6) rejects $T$. $T$ is positive and trace-preserving yet not CP, so it is no channel — but the partial transpose it defines is exactly the Peres–Horodecki entanglement test.

3 · How much a depolarizing channel shrinks the Bloch sphere

The single-qubit depolarizing channel is $\mathcal{E}(\rho)=(1-p)\rho+\tfrac{p}{3}(X\rho X+Y\rho Y+Z\rho Z)$. Writing $\rho=\tfrac12(I+\vec r\cdot\vec\sigma)$, find how the Bloch vector $\vec r$ transforms, and evaluate at $p=0.3$.

Step 1. Each Pauli conjugation flips the sign of the two components orthogonal to its axis, so $X\rho X+Y\rho Y+Z\rho Z=\tfrac12(3I-\vec r\cdot\vec\sigma)$. The three-term sum triples $I$ but reverses $\vec r$.

Step 2. Assemble: $\mathcal{E}(\rho)=\tfrac12\big[(1-p)(I+\vec r\cdot\vec\sigma)+\tfrac{p}{3}(3I-\vec r\cdot\vec\sigma)\big]=\tfrac12\big[I+(1-\tfrac{4p}{3})\,\vec r\cdot\vec\sigma\big]$, so $\vec r\mapsto(1-\tfrac{4p}{3})\,\vec r$. The channel shrinks every Bloch vector isotropically by $1-4p/3$.

Step 3. At $p=0.3$ the factor is $0.6$, so a length-$0.6$ vector maps to $0.36$. At $p=\tfrac34$ it is $0$ — the channel becomes completely depolarizing and erases all information, the third row of the limiting-cases table. At $p=0.3$ the sphere shrinks to 60% of its radius; $p=3/4$ collapses it to the centre.

The theory of quantum channels was assembled between the mid-1950s and mid-1970s by mathematicians studying positive maps on operator algebras, before physicists had a name for the objects. Its keystone — that the elusive "complete positivity" is nothing more than a positive matrix — belongs to one person.

Texts. Nielsen & Chuang, Quantum Computation and Quantum Information, Ch. 8 (quantum noise and quantum operations) · Preskill, Lecture Notes for Physics 219, Ch. 3 (measurement and evolution) · Watrous, The Theory of Quantum Information, Ch. 2 (linear maps and channels) · Wilde, Quantum Information Theory, Ch. 4 (the noisy quantum channel) and Ch. 11 (quantum entropy and data processing) · Kitaev, Shen & Vyalyi, Classical and Quantum Computation, section on superoperators · Choi, Completely Positive Linear Maps on Complex Matrices, Lin. Alg. Appl. 10 (1975) 285.

Provenance. The CPTP definition and the three representations follow Nielsen & Chuang Ch. 8, Preskill Ch. 3, and Watrous Ch. 2. A verifier should recompute: (1) the transpose counterexample — partially transposed Bell state eigenvalues $\{\tfrac12,\tfrac12,\tfrac12,-\tfrac12\}$ and $J(T)=\text{SWAP}$ eigenvalues $\{+1,+1,+1,-1\}$; (2) the amplitude-damping Choi spectrum $\{2-\gamma,\gamma,0,0\}$, giving $\{1.8,0.2,0,0\}$ at $\gamma=0.2$ with $\mathrm{Tr}_B J=I$; (3) the depolarizing Bloch shrink $1-\tfrac{4p}{3}$, equal to $0.6$ at $p=0.3$ and $0$ at $p=\tfrac34$. The data-processing proof reduces to Lindblad–Uhlmann monotonicity (Wilde Ch. 11). All quoted numbers were independently reproduced numerically for this article.

Builds on: Density operators and the trace · Tensor products and partial trace · Von Neumann Entropy · Quantum Relative Entropy. Leads to: Kraus Representation of Quantum Channels · Stinespring Dilation Theorem · Choi–Jamiołkowski Isomorphism · Complementary Channels and Degradability · Data-Processing Inequality · Holevo Bound.

Ask the tutor: Want to see the amplitude-damping Choi matrix diagonalized entry by entry, or watch the data-processing inequality collapse three states through a noisy channel — just ask.